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Differences Between Robin and Neumann Eigenvalues.

Zeév Rudnick1, Igor Wigman2, Nadav Yesha3

  • 1School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.

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This study analyzes Robin-Neumann gaps for planar domains, revealing a limiting mean value. For ergodic billiards, these gaps converge to the mean along a density-one subsequence.

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Area of Science:

  • Mathematical Physics
  • Partial Differential Equations
  • Spectral Theory

Background:

  • The study concerns boundary value problems on bounded planar domains.
  • Eigenvalues and spectral properties are central to understanding the behavior of solutions.

Purpose of the Study:

  • To investigate the Robin-Neumann gaps for a bounded planar domain with a piecewise smooth boundary.
  • To determine the limiting behavior and bounds of these gaps for various classes of domains.

Main Methods:

  • Analysis of the Robin boundary value problem and its associated eigenvalues.
  • Investigation of spectral gaps, specifically the Robin-Neumann gaps.
  • Asymptotic analysis for different domain types, including ergodic billiards, rectangles, and disks.

Main Results:

  • A limiting mean value for Robin-Neumann gaps is established for a wide class of planar domains.
  • For smooth domains, an upper bound of and a uniform lower bound are derived.
  • For ergodic billiards, convergence to the mean value is shown along a density-one subsequence.
  • Specific uniform upper bounds are found for rectangles, and improved upper bounds for disks.

Conclusions:

  • The Robin-Neumann gaps exhibit predictable behavior, converging to a mean value in many cases.
  • The study provides valuable insights into the spectral properties of planar domains with different boundary conditions.
  • Specific geometric shapes like rectangles and disks show distinct gap properties.